Photonic realization of the deformed Dirac equation via the segmented graphene nanoribbons under inhomogeneous strain

TL;DR

This paper demonstrates how engineered optical waveguide arrays can simulate a deformed Dirac equation and graphene nanoribbons under strain, enabling the study of relativistic quantum phenomena like Zitterbewegung in a controllable optical system.

Contribution

It introduces a novel optical analogue platform using waveguide arrays to simulate deformed Dirac equations and strained graphene nanoribbons, facilitating experimental exploration of relativistic effects.

Findings

Amplitude of Zitterbewegung oscillations varies with deformation parameter

Optical system accurately models strained graphene nanoribbons

Simulation provides insights into relativistic quantum phenomena

Abstract

Starting from an engineered periodic optical structure formed by waveguide arrays comprised of two interleaved lattices, we simulate a deformed Dirac equation. We show that the system also simulate graphene nano ribbons under strain. This optical analogue allows us to study the phenomenon of Zitterbewegung for the modified Dirac equation. Our results show that the amplitude of Zitterbewegung oscillations changes as the deformation parameter is changed.

Equations

$$H\psi =-i\hslash v_f\alpha \frac{\partial \psi }{\partial x}+\varphi (x)\beta \psi$$

$$H\psi =-i\hslash v_{f}f(x)\alpha \frac{\partial \psi }{\partial x}+{\varphi }_0\beta \psi$$

$$-i\frac{d{E}_n}{dz}=(-1{)}^n\beta E_n+c_{n,n-1}{E}_{n-1}+c_{n,n+1}{E}_{n+1},$$

$$i\frac{d{E}_{2n}}{dz}+c_{1}f(2n+1)E_{2n+1}+c_{1}f(2n)E_{2n-1}-\beta E_{2n}=0,$$

$$i\frac{d{E}_{2n-1}}{dz}+c_{1}f(2n)E_{2n}+c_{1}f(2n-1)E_{2n-2}+\beta E_{2n-1}=0.$$

$$i\frac{d{\psi }_1(2n,z)}{dz}+i{c}_{1}f(2n+1){\psi }_2(2n+1,z)-i{c}_{1}f(2n){\psi }_2(2n,z)-\beta {\psi }_1(2n,z)=0,$$

$$i\frac{d{\psi }_2(2n,z)}{dz}+i{c}_{1}f(2n){\psi }_1(2n,z)-i{c}_{1}f(2n-1){\psi }_1(2n-1,z)+\beta {\psi }_2(2n,z)=0.$$

$$f(n)=\sqrt{1+\frac{{\chi }_a\mu }{\nu }(1-n)},$$

$$i\frac{d{\psi }_1(2n,z)}{dz}+c_{1}f(2n)(i{\psi }_2(2n+1,z)-i{\psi }_2(2n,z))-\beta {\psi }_1(2n,z)=0,$$

$$i\frac{d{\psi }_2(2n,z)}{dz}+c_{1}f(2n)(i{\psi }_1(2n,z)-i{\psi }_1(2n-1,z))+\beta {\psi }_2(2n,z)=0.$$

$$i\frac{d{\psi }_1(2n,z)}{dz}=-i{c}_{1}f(2n)\frac{\partial {\psi }_2(2n,z)}{\partial n}+\beta {\psi }_1(2n,z),$$

$$i\frac{d{\psi }_2(2n,z)}{dz}=-i{c}_{1}f(2n)\frac{\partial {\psi }_1(2n,z)}{\partial n}-\beta {\psi }_2(2n,z).$$

$$i\frac{d\psi }{dz}=c_{1}f(x)p_x{\sigma }_x\psi +\beta {\sigma }_z\psi =c_1{P}_x{\sigma }_x\psi +\beta {\sigma }_z\psi ,$$

$$P_x=\frac{1}{\sqrt{m(x)}}{p}_x.$$

$$g(x)=\frac{1}{f^2(x)}.$$

$${\psi }_1(k,z)=G(k)[cos(\epsilon z)-i(c_{1}k+\beta )\frac{sin(\epsilon z)}{\epsilon }],$$

$${\psi }_2(k,z)=G(k)[cos(\epsilon z)+i(-c_{1}k+\beta )\frac{sin(\epsilon z)}{\epsilon }],$$

$$<x>_{z}=<x>_{0}+{{\mu\pi\chi_{a}\over\nu}}\int{G^{*}(k)\partial^{2}G(k)dk}-{\mu\pi{c_{1}}^{2}\chi_{a}\over 2\nu}\bigg{\{}\int{2|G(k)|^{2}({(\varepsilon^{2}z^{2}-2)\over\varepsilon^{3}})dk}+\\ \int{2|G(k)|^{2}({(\varepsilon^{2}z+ikc_{1})\over\varepsilon^{4}})sin(2\varepsilon z)dk}-\int{|G(k)|^{2}({4(ikzc_{1}-1)\over\varepsilon^{3}})cos^{2}(\varepsilon z)dk}\bigg{\}}.\hskip 120.00018pt$$

Full text

I Introduction

During the last decades, classical analogues of quantum and quantum-relativistic systems have attracted much attention a0 ; a00 ; a000 . There have been a significant movement towards simulating quantum phenomena in optical systems a1 ; a2 ; a3 ; a4 ; a5 ; a6 ; a7 ; a8 ; a9 ; a10 . These studies have eventually led to a new class of analogue between classical optics and quantum mechanics, such as Aharonov-Bohm effect, quantum collapses and revivals and the Berry phase, etc a3 ; a4 ; a5 . Furthermore, a series of papers on classical simulation of relativistic quantum mechanics in the optical systems have appeared including experimental realizations a7 ; a13 ; a16 ; a14 ; Keil ; Koke . One powerful method by which relativistic quantum phenomena are generated in the laboratory is photonic waveguides a016 .

Graphene is an ideal candidate for simulating the Dirac equation in the lab. It can be imagined as a layer of carbon atoms and its electrons can be regarded as relativistic particles. Despite many properties of graphene, the absence of a band gap is the biggest obstacle to be used in electronic devices g1 ; g2 ; g22 . In recent years, much effort has been made on engineering the band gap by applying strain. In this case, charge carriers obey a generalized Dirac equationg3 ; g4 ; g5 .

Photonic waveguides play a significant role in simulating relativistic quantum physics g6 . Indeed, one dimensional Dirac equation can be realized by spatial beam propagation in binary waveguide arrays composed of two type of equally spaced waveguides a16 . It was shown that this setup can be described by a simplified system of coupled mode equations that allows us to obtain the standard Dirac equation under certain assumptions. In addition, photonic waveguides can be engineered to simulate quantum phenomena a14 ; a014 ; a0014 ; a00014 and to realize the non-linear coherent states a0016 ; a316 ; a416 . It is important to emphasize that coherent states are extremely useful in physics. The concept of coherent states has been developed in many different branches of physics such as mathematical physics a015 and quantum optics a0015 . Recently, the coherent states of deformed Heisenberg-Weyl algebra have been investigated a15 by using a waveguide lattices with specific coupling coefficients between them.

The motivations to study deformations are manifolda20 ; a21 ; b21 . One well-known deformation occurs due to the so-called doubly special relativity which proposes that both the velocity of light, the Planck energy are universal constants a22 . The deformation can also emerge due to the existence of a minimum measurable length a022 . Another famous deformations are caused by position-dependent mass where effective mass depend on the position mm1 ; mm2 ; mm3 ; mm4 , which is used in semiconductor heterostructures mm5 , quantum dots mm6 , semiconductor theory mm7 and problems in condensed-matter physics mm8 . Deformed canonical commutations are obtained by modifying the metric structure associated with a curved space mm9 ; mm10 ; mm11 . In this approach there is a causal relationship between deforming function and the metric tensor mm12 ; mm13 .

In this work, we propose a 1D periodic array of coupled waveguides in which separations between the waveguides are controlled in order to simulate deformed Dirac equation. We first construct the deformed relativistic wave equation making use of deformed Lie algebras. This deformation plays the role of nonlinearities in our model. We then discuss Zitterbewegung (ZB), an extremely fast oscillation of relativistic particles. We compare the ZB effect in the deformed model against the original one, showing that the amplitude and frequency change with the deformation parameter. The paper is organized as follows. In Sec. II, we introduce segmented graphene nano ribbons under strain which provides an experimental tool to realize a generalized Dirac equation. We then show how the same model arises in engineered photonic waveguides in Sec. III, and study the ZB in the deformed scenario in Sec. IV. We conclude in Sec. V.

II Segmented graphene nanoribbons under strain

Graphene is a single atomic layer of carbon arranged in a honeycomb lattice. Among the carbon nanostructures, graphene Nanoribbons(GNR) that are narrow strips of graphene have garnered great interest in recent years. As shown in Ref. b24 for narrow GNRs we have the following 1D generalized Dirac equation:

[TABLE]

where $\phi(x)$ is position dependent scalar Higgs field. Here we choose $\phi(x)\equiv\phi_{0}$, in which case the equation becomes the usual 1D Dirac equation b25 . The effects of nonisotropic strain on GNR can be summarised by the substitution $p_{x}\rightarrow f(x)p_{x}$ in the above Hamiltonian b26 :

[TABLE]

We will show that this deformed Dirac equation can also be simulated by a specifically engineered waveguide array.

III Simulation of the deformed Dirac equation by binary photonic superlattices

Within the nearest-neighbour coupling approximation the propagation of an optical field in disordered waveguide arrays, comprised of two interleaved lattices A and B, is described by the following equations 1 :

[TABLE]

where $E_{n}$ is the electric field amplitude at the nth waveguide ($n=1,2,...,N$), $\beta$ is the propagation constant and $c_{n,n\pm 1}$ are the coupling coefficients between waveguides. As we show in Fig. (1) the coupling constant depends on the separation distance 001 . Thus we can control the coupling coefficients, by controlling the separations between waveguide elements. Here we choose the coupling coefficients as: $c_{n,n-1}=c_{1}{f(n)}$ at any site $n>1$ a15 . Consequently, the coupled-mode equations take the following form 2 :

[TABLE]

and

[TABLE]

By setting $E_{2n}=(-1)^{n}\psi_{1}(2n,z)$, $E_{2n-1}=-i(-1)^{n}\psi_{2}(2n-1,z)$, Eqs. (4) and (5) become

[TABLE]

and

[TABLE]

We consider the deformation 4

[TABLE]

in which $\chi_{a}$ is an anharmonicity parameter ($0\leq\chi_{a}\leq\nu$ ) and $\mu=\pm 1$. To achieve this in our system we choose $c_{n,n-1}=c_{1}e^{-d_{(n,n-1)}+\kappa d_{1}\over{d_{1}}}$, where $d_{1}$ is the distance between the first coupled waveguides and $\kappa$ is a positive constant. By choosing ${\chi_{a}\over\nu}={1\over 100}$, one can make an approximation $f(2n\pm 1)\simeq f(2n)$ which s necessary in order for our system to mimic the relativistic wave equation. Indeed, we can rewrite Eqs. (6) and (7) as

[TABLE]

and

[TABLE]

Finally, by employing $\psi(n\pm 1,z)=\psi(n,z)\pm{\partial\psi(n,z)\over\partial n}$, the coupled-mode equations can be written in the following form:

[TABLE]

and

[TABLE]

Therefore, making a replacement $n\rightarrow x$ and substituting Eq. (8) into Eqs. (11) and (12) yields

[TABLE]

where $\psi=(\psi_{1}(x,z),\psi_{2}(x,z))^{\dagger}$ and $P_{x}=-i\partial_{X}=-if(x)\partial_{x}$ with $X=-{2\mu\nu\over\chi_{a}}\sqrt{1+{\chi_{a}\mu\over\nu}(1-x)}$. The above expression, after making formal changes $z\rightarrow t$, $\beta\rightarrow m$, $c_{1}\rightarrow c$ and setting $\hbar=c=1$ , reduces to the sought-for deformed Dirac equation. Notice that we can interpret the above deformed momentum in terms of position-dependent-mass formalism. In this case, mass is a function of the position as follows p0 ; p1 :

[TABLE]

Another way to interpret the deformed momentum in Eq. (13) is in terms of the Dirac equation in a curved space p2 ; p3 ; p4 ; p5 ; p6 with the metric p7

[TABLE]

IV Zitterbewegung in the deformed model

The trembling motion of relativistic particles caused by an interference between positive and negative energy wave components is known as Zitterbewegung m4 ; m04 ; m004 . To simulate ZB in our waveguide setup the elements in the array are excited by a broad beam $E(x,0)$ with the propagating field envelopes $G(X)$. We assume that $G(X)$ change slowly over the waveguide spacing, i.e., $G(c_{1}f(2n))\simeq G(c_{1}f(2n-1))$. The solution to Eq. (13) in the momentum space reads

[TABLE]

and

[TABLE]

where $\varepsilon=\sqrt{{c_{1}}^{2}k^{2}+{\beta}^{2}}$ and $G(k)={1\over 2\pi}\int dXG(X)e^{-ikX}$. After some straightforward calculations, the average position in the array becomes

[TABLE]

The last two terms in the above equation are oscillatory terms that yields ZB as shown in Fig. (2). The amplitude of oscillation changes because of the different periodicity in the x direction($\mu=1$ and $\mu=-1$ which means as n increase the separation distance between waveguides decrease and increase respectively).

In our system $\beta$ is the propagation constant and plays the role of mass in the deformed Dirac equation. Figures 2(a) and (b) show that the amplitude of oscillation decreases while the frequency increases as the mass is increased. Changes in ZB are different as one varies the deformation parameter as depicted in Figs. 2(c) and (d). The amplitude of ZB increases while the frequency stays the same, as the deformation parameter is increased. Note that, in the limit ${\chi_{a}\over\nu}\rightarrow 0$, one gets back the usual model which there is not any change in the amplitude of oscillations.

V Conclusions

We have derived a coupled equations for light propagation in the waveguide lattices which is comprised of two interleaved lattices with specifically engineered coupling coefficients between neighbouring waveguides. We showed that the resulting equation is equivalent to a deformed Dirac equation that arises in graphene nano ribbons under strain and also showed a connection between the deformed Dirac equation with position-dependent mass and curved space. Lastly, we have calculated the average position in the deformed model and found that ZB changes as the deformation parameter is changed.